The present embodiments relate to a method and an apparatus for determining attenuation coefficients for an object using a movable x-ray source and a detector.
X-ray methods are standard methods in medical technology. In the case of simple x-ray recordings, x-ray radiation is transmitted through an object to be examined and is then recorded by a detector. The recording or projection represents an item of information about the attenuation of the transmitted x-rays on a path through the object. The attenuation of the x-ray radiation depends on the density of the parts of the object, through which the x-ray radiation is radiated. The density supplies information about the nature of the object, which may be shown in visual form for diagnoses. For an x-ray recording, the intensity registered by the detector depends on the overall composition of the object on the line along which the x-rays pass (e.g., data integrated over a line is available). Attenuation coefficients may not be obtained for the object from one x-ray recording as a function of all three spatial coordinates. For a three-dimensional image, a plurality of x-ray recordings is made from different recording positions. From the plurality of x-ray recordings, a three-dimensional image is reconstructed. One of the most important technologies in medical technology that takes this approach is computer tomography (CT). As part of the CT, the x-ray source and the x-ray detector traverse a path or trajectory. In doing this, recordings are made along the trajectory. From these recordings, a three-dimensional image is reconstructed from attenuation coefficients that are associated with the density.
Image reconstruction in transmission computer tomography is a complex mathematical problem. For the construction of three-dimensional images, two groups of methods have become established: exact methods (cf. Katsevich, A., “Theoretically Exact FBP-Type Inversion Algorithm for Spiral CT,” SIAM J. Appl. Math., Vol. 62, No. 6 (2002): pp. 2012-26; Katsevich, A., “Image Reconstruction for the Circle-and-Line Trajectory,” Phys. Med. Biol., Vol. 49, No. 22 (2004): pp. 5059-72; Katsevich, A., “Image Reconstruction for the Circle-and-Arc Trajectory,” Phys. Med. Biol., Vol. 50, No. 10 (2005): pp. 2249-65; and Pack, J. and F. Noo, “Cone-Beam Reconstruction Using 1D Filtering Along the Projection of M-Lines,” Inverse Problems, Vol. 21, No. 3 (2005): pp. 1105-20); and approximative methods (cf. Yu, H. and G. Wang “Feldkamp-type VOI reconstruction from super-short-scan cone-beam data,” Med. Phys., Vol. 31, No. 6 (2004): pp. 1357-62). These may be (theoretically) exact methods that contain no mathematical approximations; the numeric implementation and the technical realization may, however, involve errors.
These methods calculate the 3D density distribution of the object under examination from 2D projection data, essentially taking into account the following acts: (i) calculation of the numeric derivative of the projections recorded along the sample path of the x-ray source (see Noo, F., et al., “A New Scheme for View-Dependent Data Differentiation in Fan-Beam and Cone-Beam Computed Tomography,” Phys. Med. Biol., Vol. 52, No. 17 (2007): pp. 5393-414 for various possibilities), (ii) 1-D displacement-invariant filtering of the differentiated projection data along a family of filter lines, and (iii) weighted back-projection of the filtered projections into the image volume.
In practical applications, the projection data is not available in continuous form, but in discrete form because, the result of the data recording is a finite number of projection images, each of which is available in sampled form. During the reconstruction, interpolation steps therefore occur. The interpolations may have a negative effect on the quality of the resulting reconstructed image (e.g., in that the interpolations limit the maximum achievable spatial resolution).
Until now, reconstruction methods have been implemented such that interpolation operations are performed during the calculation of the derivatives, the filtering and the back-projection. Between the individual calculation steps, the results are held in temporary storage. The calculation of the numeric derivative is performed such that the results are obtained at the original detector positions, even though the derivatives are required at other positions, determined by the filtering lines during the filtering. An interpolation is used if the derivative is calculated on a Cartesian grid, but the filtering lines are not parallel to the axes of the grid. In this case, the filtering lines for the Feldkamp method discussed in “Feldkamp-type VOI reconstruction from super-short-scan cone-beam data” run along horizontal lines in the x-ray image detector. With the newer approximative and exact reconstruction methods discussed in the references above, in conjunction with new types of sampling paths such as, for example, circle-and-line, circle-and-arc and saddle, the filtering lines used are mostly non-horizontal.
Interpolation may be used both in the calculation of the derivative and also in the extraction of the filtering lines on the projection images. The extraction of the filtering lines may be critical for the image quality (e.g., spatial resolution), for which reason improved interpolation methods have been proposed (Joseph's method; see Noo, F., et al., “Exact helical reconstruction using native cone-beam geometrics,” Phys. Med. Biol., Vol. 48, No. 23 (2003): pp. 3787-818) in order to minimize the loss of image quality.